Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows

Abstract

Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold \(M\) admitting an Anosov flow must have exponential growth, and \(M\) must be universally covered by \(\mathbb{R}^{3}\). Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold $M_n$ that admits $n$ pairwise inequivalent Anosov flows for any natural number $n$. In all known examples, the manifolds $M_n$ are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds $M_{k,n}$, $k \in \mathbb{N}$ with a single geometric piece, each of which admits $n$ pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.